Lecture 12: Integral representation of the basic
hypergeometric function q -Integration by parts.
Aims:
Finish our discussion of q -integrals
Prove the q -integral representation of the basic hyperg
Lecture 19: More q -Ladder operators.
Aims:
To nd the linearly independent solution to the q -dierence equation in
terms of the associated function.
19.1. q -dierence-dierence equations for orthogona
Lecture 20: Example:q -Hermite Polynomials
Aims:
To get a grip on the mechanics of the developed theory by detailing an
example.
Last lecture we showed that the discrete Hermite polynomials, dened by
Lecture 23: Singularity connement.
Aims:
Dene singularity connement for a mapping
Let us consider the QRT mapping arising from the integral
In =
1
1
1
+
+ xn1 + xn +
xn
xn1
xn xn1
so we calculate, In
Lecture 21: q -Airy and q -Bessel functions.
Aims:
To introduce the q -analogues of the Airy and Bessel functions
20.1. q -Airy functions. Let us start with the Airy functions. The Airy function is a
Lecture 15:More denitions and examples.
Aims:
To discuss the three term recurrence relation further for orthogonal polynomils.
Present more examples that motivate the presented theory.
Last lecture,
Lecture 18: q -Ladder operators.
Aims:
To determine the q -dierence equation satised by orthogonal polynomials.
18.1. q -dierence equations for orthogonal polynomials. The aim of
this section is to n
Lecture 14: Orthogonal polynomials.
Aims:
Give some sort of preface to the courses treatment of orthogonal polynomials
Describe some of the common elements to orthogonal polynomials and
q -orthogona
Lecture 17: More ladder operators.
Aims:
To complete our description of the linear problem for orthogonal polynomial ensembles
Provide the linearly independent
Furnish our work with a simple exampl
Outline of the course
I Discrete Calculus :
q -Derivatives and h-Derivatives.
Discrete product rules and quotient rules.
Discrete Taylor expansions.
The two q -exponential functions.
q -Trigonome
Lecture 2: Taylors Formula.
Aims:
Present a generalized form of Taylors Theorem.
2.1. Taylors Theorem. Let us consider a function, f , which possesses all
its derivatives at a point a. Such a functio
Lecture 5: Binomial identities.
Aims:
To give some analogous q -binomial identities.
To present a combinatorial application of the q -calculus.
5.1. Binomial Identities. Last lecture we proved two v
Lecture 11: The q -integral.
Aims:
Discuss the general q -antiderivative/integral.
Introduce Jacksons denite q -integral.
11.1. Antiderivative. Some functions are naturally expressed in terms of
int
Lecture 13: q -Gamma q -Beta functions.
Aims:
To describe two q -functions, the q -Gamma and q -Beta functions, in terms
of integrals
To describe the q -logarithm.
13.1. Gamma and Beta functions. Th
Lecture 7: q -exponential functions.
Aims:
Consider the innite products (1 + x) .
q
Introduce the two analogues of the q -exponential function.
7.1. The q -exponential functions and (1 + x) . We wis
Lecture 3: Taylors formula continued.
Aims:
To continue our discussion of Taylor polynomials.
Demonstrate some basic identities.
3.1. The q -Taylor polynomials. So last lesson, the polynomials speci
Lecture 4: q -Binomials.
Aims:
To introduce the q -Binomials
Introduce some (non-assessable) quantum systems.
4.1. q -Binomials. We have shown that the sequence of polynomials, given by
Pn (x) =
(x
Lecture 16: Ladder operators.
Aims:
Itroduce the determine a dierential equation satised by an orthogonal
polynomial ensemble.
A topic that is close to my heart (I have a fair few results in this dir